I cannot understand some part of the period signal's Fourier transform.

Here this my note's methods,

For periodic signal with period $T_0$, define as $s_{T_0}(t)$ as $$ s_{T_0} (t) = \begin{cases} s(t) for -T_0/2 <t<T_0/2\\ 0 & \text{otherwise.} \end{cases} $$ $$s(t)=\sum_{n=-\infty}^{\infty}s_{T_0}(t-n T_0) = s_{T_0}(t) * \sum_{n=-\infty}^{\infty} \delta (t-n T_0)$$ So the Fourier transform will be $$S(f)=S_{T_0}(f) \cdot f_0 \sum_{n=-\infty}^{\infty}\delta(f-n f_0) = f_0 \sum_{n=-\infty}^{\infty}S_{T_0}(n f_0) \delta(f-n f_0)$$ and the other methods is $$x(t)= s(t) \cdot \sum_{n=-\infty}^{\infty} \delta (t-n T_0)$$ $$X(f)=S(f) * f_0 \sum_{n=-\infty}^{\infty}\delta(f- n f_0) = f_0 \sum_{n=-\infty}^{\infty}S(f-n f_0)$$

I cannot understand why I have "$\cdot f_0$" after Fourier Transform and why "$\delta(f-n f_0)$" can transform to "$\delta(t-nT_0)$"? And the last questions is, for the periods signal, why the note can use Fourier transform directly with out considering the Fourier serials?

I cannot understand some part of the period signal's Fourier transform.

Here this my note's methods,

For periodic signal with period $T_0$, define as $s_{T_0}(t)$ as $$ s_{T_0} (t) = \begin{cases} s(t) for -T_0/2 <t<T_0/2\\ 0 & \text{otherwise.} \end{cases} $$ $$s(t)=\sum_{n=-\infty}^{\infty}s_{T_0}(t-n T_0) = s_{T_0}(t) * \sum_{n=-\infty}^{\infty} \delta (t-n T_0)$$ So the Fourier transform will be $$S(f)=S_{T_0}(f) \cdot f_0 \sum_{n=-\infty}^{\infty}\delta(f-n f_0) = f_0 \sum_{n=-\infty}^{\infty}S_{T_0}(n f_0) \delta(f-n f_0)$$ and the other methods is $$x(t)= s(t) \cdot \sum_{n=-\infty}^{\infty} \delta (t-n T_0)$$ $$X(f)=S(f) * f_0 \sum_{n=-\infty}^{\infty}\delta(f- n f_0) = f_0 \sum_{n=-\infty}^{\infty}S(f-n f_0)$$

I cannot understand why I have "$\cdot f_0$" after Fourier Transform and why "$\delta(f-n f_0)$" can transform to "$\delta(t-nT_0)$"? And the last questions is, for the periods signal, why the note can use Fourier transform directly with out considering the Fourier serials?

I cannot understand some part of the period signal's Fourier transform.

Here this my note's methods,

For periodic signal with period $T_0$, define as $s_{T_0}(t)$ as

$$ s_{T_0} (t) = \begin{cases} s(t) for -T_0/2 <t<T_0/2\\ 0 & \text{otherwise.} \end{cases} $$

$$s(t)=\sum_{n=-\infty}^{\infty}s_{T_0}(t-n T_0) = s_{T_0}(t) * \sum_{n=-\infty}^{\infty} \delta (t-n T_0)$$ So the fourier transform will be $$S(f)=S_{T_0}(f) \cdot f_0 \sum_{n=-\infty}^{\infty}\delta(f-n f_0) = f_0 \sum_{n=-\infty}^{\infty}S_{T_0}(n f_0) \delta(f-n f_0)$$

and the other methods is

$$x(t)= s(t) \cdot \sum_{n=-\infty}^{\infty} \delta (t-n T_0)$$ $$X(f)=S(f) * f_0 \sum_{n=-\infty}^{\infty}\delta(f- n f_0) = f_0 \sum_{n=-\infty}^{\infty}S(f-n f_0)$$

I cannot understand why I have "$\cdot f_0$" after Fourier Transform and why "$\delta(f-n f_0)$" can transform to "$\delta(t-nT_0)$"? And the last questions is, for the periods signal, why the note can use fourier transform directly with out considering the fourier serials?

I cannot understand some part of the period signal's Fourier transform.

Here this my note's methods,

For periodic signal with period $T_0$, define as $s_{T_0}(t)$ as $$ s_{T_0} (t) = \begin{cases} s(t) for -T_0/2 <t<T_0/2\\ 0 & \text{otherwise.} \end{cases} $$ $$s(t)=\sum_{n=-\infty}^{\infty}s_{T_0}(t-n T_0) = s_{T_0}(t) * \sum_{n=-\infty}^{\infty} \delta (t-n T_0)$$ So the Fourier transform will be $$S(f)=S_{T_0}(f) \cdot f_0 \sum_{n=-\infty}^{\infty}\delta(f-n f_0) = f_0 \sum_{n=-\infty}^{\infty}S_{T_0}(n f_0) \delta(f-n f_0)$$ and the other methods is $$x(t)= s(t) \cdot \sum_{n=-\infty}^{\infty} \delta (t-n T_0)$$ $$X(f)=S(f) * f_0 \sum_{n=-\infty}^{\infty}\delta(f- n f_0) = f_0 \sum_{n=-\infty}^{\infty}S(f-n f_0)$$

I cannot understand why I have "$\cdot f_0$" after Fourier Transform and why "$\delta(f-n f_0)$" can transform to "$\delta(t-nT_0)$"? And the last questions is, for the periods signal, why the note can use Fourier transform directly with out considering the Fourier serials?